Finite Group Databasebeta

← #29707361

Subgroups — #29707361

Order 15616.

1250 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840390C211yesmaximal, characteristic, complemented
2C2×#16839763C211yesmaximal, characteristic
3#21448411C211yesmaximal, characteristic
4#21446968C212yesmaximal, complemented
5#21446963C212yesmaximal, complemented
6#21446958C212yesmaximal, complemented
7#21446953C212yesmaximal, complemented
8#21445540C211yesmaximal, characteristic, complemented
9C2×#16839525C211yesmaximal, characteristic, complemented
10#21445515C211yesmaximal, characteristic, complemented
11C2×C61×#416C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13#16840390—24no—
14#16840390—24no—
15C2×#14519777—21no—
16C2×#14519776C2212yes—
17C2×#14519769—21no—
18C2×#14519768C2212yes—
19C2×#14519754—22no—
20C2×#14519736C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22C22×C4⋊Dic61C2211yescharacteristic
23#16839763—24no—
24C2×#14519701C2212yes—
25C2×#14519698C2212yes—
26#16839669C2212yes—
27#16839665C2212yes—
28#16839642—22no—
29#16839622—21no—
30#16839622—21no—
31#16839525C2212yes—
32#16839525C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35#16839516C2212yes—
36#16839516C2212yes—
37C22×#41×C61C2211yescharacteristic
38C23×C8×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C61×#416—24no—
41C2×#103×C61C2211yescharacteristic
42C2×#103×C61C2211yescharacteristic
43C2×#102×C61C2211yescharacteristic
44C2×#102×C61C2211yescharacteristic
45C22×Dic122—22no—
46C23×Dic61—21no—
47C23×Dic61—21no—
48C2×#11758194—22no—
49C2×#11758194—22no—
50C2×#11758194—22no—
51C22×#92418C2312yes—
52#14519777—28no—
53#14519776—24no—
54#14519776—24no—
55#14519769—28no—
56#14519768—24no—
57#14519768—24no—
58#14519754—216no—
59#14519736—22no—
60#14519736—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61D414yes—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—22no—
73C2×#11758196—22no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—22no—
76C2×C22⋊Dic61D412yes—
77C2×C4×Dic61C2312yes—
78#14519701—24no—
79#14519698—24no—
80#14519684—22no—
81#14519684—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61Q814yescomplemented
88C22×Q8×C61C2311yescharacteristic
89C22×D4×C61—21no—
90C22×D4×C61C2311yescharacteristic
91C23×C4×C61—21no—
92C23×C4×C61C2311yescharacteristic
93C22×C8×C61—22no—
94C22×C8×C61—22no—
95C22×C8×C61—22no—
96C22×C8×C61C2311yescharacteristic
97C22×C8×C61C2311yescharacteristic
98#117×C61—22no—
99#117×C61D414yes—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61—21no—
104#103×C61—22no—
105#103×C61—22no—
106#102×C61—22no—
107#102×C61—22no—
108C24×C61—21no—
109C24×C61C2×D411yescharacteristic
110#41×C61—28no—
111#41×C61—28no—
112#41×C61—44no—
113#41×C61—44no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61C2×Q812yes—
120C2×D4×C61—24no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—42no—
127C2×D4×C61—42no—
128C2×D4×C61C2×Q812yes—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—24no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×Q812yes—
139C22×C4×C61#4112yes—
140C22×C4×C61#4112yes—
141C22×C4×C61#4112yes—
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—22no—
144C22×Dic61—24no—
145C22×Dic61—22no—
146C22×Dic61—42no—
147C22×Dic61—81no—
148C22×Dic61—81no—
149C22×Dic61—81no—
150C22×Dic61—81no—
151C22×Dic61C2×D412yes—
152C22×Dic61C2×D412yes—
153C2×Dic122—28no—
154C2×Dic122—82no—
155C2×Dic122—82no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—44no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C2×D412yes—
166C4⋊C4×C61C2×D412yes—
167C22⋊C4×C61—24no—
168C42×C61—24no—
169C22⋊Dic61—24no—
170C22⋊Dic61—82no—
171C22⋊Dic61—82no—
172C4⋊Dic61—28no—
173C4⋊Dic61—24no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—42no—
180C4⋊Dic61C2×D414yes—
181#11758196—28no—
182#11758196—28no—
183C4×Dic61—22no—
184C4×Dic61—22no—
185#11758194—44no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—22no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61#12012yes—
201C23×C61C22⋊Q812yes—
202C23×C61#12412yes—
203C23×C61C22×D411yescharacteristic
204Q8×C61—22no—
205Q8×C61—22no—
206Q8×C61—41no—
207Q8×C61—41no—
208D4×C61—22no—
209D4×C61—22no—
210D4×C61—48no—
211D4×C61—41no—
212D4×C61—41no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—48no—
223C2×C4×C61#12012yes—
224C2×C4×C61C22⋊Q814yes—
225C2×C4×C61C22⋊Q814yes—
226C2×C4×C61C22⋊Q812yes—
227C2×C4×C61#12414yes—
228C2×C4×C61#12414yes—
229C2×C4×C61#12412yes—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×Q811yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—24no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—84no—
249C2×Dic61—84no—
250Dic122—88no—
251C8×C61—41no—
252C8×C61—41no—
253#92418—42no—
254#23458—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61#39314yes—
269C22×C61C2×#12011yescharacteristic
270C22×C61C2×C22⋊Q811yescharacteristic
271C22×C61C2×#12411yescharacteristic
272C22×C61#57611yescharacteristic
273C22×C61#57711yescharacteristic
274C22×C61#57711yescharacteristic
275C22×C61#57811yescharacteristic
276C4×C61—42no—
277C4×C61—42no—
278C4×C61—42no—
279C4×C61#39314yes—
280C4×C61C2×C22⋊Q811yescharacteristic
281C4×C61C2×C22⋊Q811yescharacteristic
282C4×C61C2×#12411yescharacteristic
283C4×C61C2×#12411yescharacteristic
284Dic61—42no—
285Dic61—44no—
286Dic61—42no—
287Dic61—82no—
288C2×#544—611no—
289C2×#417—611no—
290C2×#416D6111yescharacteristic
291#2311—611no—
292#2026—612no—
293#2025—612no—
294#2024—612no—
295#2023—612no—
296#1736—611no—
297C2×#340—611no—
298#1729—611no—
299C2×C61—24no—
300C2×C61—82no—
301C2×C61C2×#39311yescharacteristic
302C2×C61#301211yescharacteristic
303C2×C61#301311yescharacteristic
304C2×C61#308311yescharacteristic
305C2×C61#308411yescharacteristic
306C2×C61#313611yescharacteristic
307C2×C61#313711yescharacteristic
308C22×#41C2×D6111yescharacteristic
309C22×M16—611no—
310C23×C8C2×D6111yescharacteristic
311#544—1224no—
312#544—1224no—
313C2×C4⋊Q8—612no—
314C2×C4⋊Q8—1221no—
315C2×#125—612no—
316C2×#125—1221no—
317C2×C22⋊Q8—1222no—
318C2×#117—611no—
319C2×#117C2×D6111yescharacteristic
320C22×C4⋊C4—611no—
321C22×C4⋊C4—611no—
322#417—1224no—
323#416—24no—
324C2×#103—612no—
325C2×#103C2×D6111yescharacteristic
326C2×#103C2×D6111yescharacteristic
327C2×#102—612no—
328C2×#102C2×D6111yescharacteristic
329C2×#102C2×D6111yescharacteristic
330#391—612no—
331#390—612no—
332#386—1222no—
333#380—1221no—
334#380—1221no—
335#340—612no—
336#340—612no—
337#340—612no—
338#340—612no—
339#336—612no—
340#336—612no—
341C61#2345811yescharacteristic, complemented, Sylow
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41Dic12214yescomplemented
348C22×Q8—1222no—
349C22×Q8C22×D6111yescharacteristic
350C22×D4—21no—
351C22×D4C22×D6111yescharacteristic
352C23×C4—21no—
353C23×C4—1221no—
354C23×C4—1221no—
355C23×C4C22×D6111yescharacteristic
356C2×M16—1222no—
357C2×M16—1222no—
358C2×M16—1222no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—22no—
362C22×C8—612no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365C4⋊Q8—1228no—
366C4⋊Q8—1224no—
367C4⋊Q8—1224no—
368#125—1228no—
369#125—1224no—
370#125—1224no—
371C22⋊Q8—12216no—
372#117—22no—
373#117—1222no—
374#117—1222no—
375#117C22⋊D6114yes—
376C2×C4⋊C4—614no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—612no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—1222no—
395C2×C42—21no—
396C2×C42—612no—
397#103—22no—
398#103—22no—
399#103—1224no—
400#102—22no—
401#102—22no—
402#102—1224no—
403#95—1222no—
404#95—1222no—
405C24—21no—
406C24D4×D6111yescharacteristic
407#41—28no—
408#41—28no—
409#41—44no—
410#41—44no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—1228no—
417C2×Q8—4882no—
418C2×Q8—4882no—
419C2×Q8C2×Dic12212yes—
420C2×D4—24no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4C2×Dic12212yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—1224no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—4881no—
441C22×C4—4881no—
442C22×C4—4881no—
443C22×C4—4881no—
444C22×C4—2442no—
445C22×C4C2×Dic12212yes—
446C22×C4#1175822212yes—
447C22×C4D4×D6112yes—
448C22×C4D4×D6111yescharacteristic
449C22×C4#1175822412yes—
450C22×C4#1175822412yes—
451C22×C4C2×C22⋊D6112yes—
452C22×C4C23×D6111yescharacteristic
453M16—2444no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—44no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—1222no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—614no—
467C4⋊C4—1228no—
468C4⋊C4—1228no—
469C4⋊C4—1228no—
470C4⋊C4—1222no—
471C4⋊C4—1224no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—2442no—
477C4⋊C4C2×C22⋊D6112yes—
478C4⋊C4C2×C22⋊D6112yes—
479C22⋊C4—24no—
480C22⋊C4—1224no—
481C22⋊C4—4882no—
482C22⋊C4—4882no—
483C42—24no—
484C42—1222no—
485C42—1222no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23#1451974512yes—
498C23#1451975412yes—
499C23#1451976312yes—
500C23C2×D4×D6111yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505Q8—4888no—
506D4—22no—
507D4—22no—
508D4—48no—
509D4—41no—
510D4—41no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—48no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1224no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—4884no—
529C2×C4—4884no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4#1451974512yes—
536C2×C4#1451975414yes—
537C2×C4#1451975412yes—
538C2×C4#1451975614yes—
539C2×C4#1451976314yes—
540C2×C4#1451976312yes—
541C2×C4#1451976414yes—
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4C22×Dic12211yescharacteristic
545C2×C4C2×#1175822411yescharacteristic
546C2×C4C2×#1175822411yescharacteristic
547C2×C4C2×#1175822211yescharacteristic
548C8—41no—
549C8—41no—
550C8—2442no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22#1683967414yes—
565C22C2×#1451974511yescharacteristic
566C22C2×#1451975411yescharacteristic
567C22C2×#1451976311yescharacteristic
568C22#1684056411yescharacteristic
569C22#1684056911yescharacteristic
570C22#1684056911yescharacteristic
571C22#1684057611yescharacteristic
572C4—42no—
573C4—42no—
574C4—42no—
575C4—4882no—
576C4—2442no—
577C4—2444no—
578C4—2442no—
579C4#1683967414yes—
580C4C2×#1451975411yescharacteristic
581C4C2×#1451975611yescharacteristic
582C4C2×#1451976311yescharacteristic
583C4C2×#1451976411yescharacteristic
584C2—24no—
585C2—82no—
586C2C2×#1683967411yescharacteristic
587C2#2145511911yescharacteristic
588C2#2145513011yescharacteristic
589C2#2145566811yescharacteristic
590C2#2145567911yescharacteristic
591C2#2145617711yescharacteristic
592C2#2145618811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenterC2×#416D61
hypocenterC61#23458
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840390C2semidirect
4. #21446968C2semidirect
5. #21446963C2semidirect
6. #21446958C2semidirect
7. #21446953C2semidirect
8. #21445540C2semidirect
9. C2×#16839525C2semidirect
10. #21445515C2semidirect
87. C2×#41×C61Q8semidirect
341. C61#23458semidirect
347. C2×#41Dic122semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168403902
C2×#168397632
#214484112
#214469682
#214469632
#214469582
#214469532
#214455402
C2×#168395252
#214455152
C2×C61×#4162
#2345861